book

Mathematics for Game Programming and Computer Graphics

by Penny de Byl

November 2022

Intermediate to advanced

444 pages

9h 53m

English

Packt Publishing

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Technical requirementsGetting Started with Python, PyCharm, and PygameCreating a Basic Graphics WindowWorking with Window and Cartesian CoordinatesLet’s do it…Your Turn…SummaryAnswers
Technical requirementsColorLet’s do it…Your turn…LinesLet’s do it…Your turn…Let’s do it…TextLet’s do it…Your turn…PolygonsLet’s do it…Your turn…Raster ImagesLet’s do it…SummaryAnswers
Technical requirementsThe Naïve Way: Drawing a line with brute forceLet’s do it…Enter Bresenham: The improved approachLet’s do it…Drawing circles the Bresenham wayLet’s do it…Anti-aliasingSummary
Technical requirementsExploring the OpenGL Graphics PipelineLet’s do it…Drawing Models with MeshesLet’s do it…Your turn…Viewing the Scene with CamerasProjecting Pixels onto the ScreenUnderstanding 3D Coordinate Systems in OpenGLPerspective ProjectionYour turn…Projecting into the NDCLet’s do it…Your turn…SummaryAnswers
Technical requirementsAdding Lighting EffectsLet’s do it…Your turn…Placing Textures on MeshesLet’s do it…Let’s do it…SummaryAnswers
Technical requirementsIntroducing the main game loopUpdating and drawing objectsLet’s do it…Your turn…Measuring timeLet’s do it…SummaryAnswers
Technical requirementsWorking with mouse interactionsLet’s do it…Your turn…Combining 2D and 3D environmentsLet’s do it…Converting the mouse to projection coordinatesLet’s do it…Your turn…Let’s do it…Adding keyboard commandsLet’s do it…Your turn…Let’s do it…SummaryAnswers

Technical requirementsComparing similar trianglesWorking with right-angled trianglesYour turn…Calculating angles with sine, cosine, and tangentInvestigating triangles by drawing 3D objectsSummaryAnswers
Technical requirementsUnderstanding the difference between points and vectorsDefining vector key operations Let’s do it…Working out a vector’s magnitude Your turn…Exploring the relationship between angles and vectorsThe dot productThe cross productSummaryAnswers
Technical requirementsDefining lines, segments, and raysUsing the parametric form of linesYour turn…Let’s do it…Calculating and displaying normalsYour turn…Let’s do it…Your turn…Summary Answers
Technical requirementsDisplaying Mesh Triangle NormalsLet’s do it…Defining polygon sides with normalsWorking with visible sides of a polygonCulling Polygons According to the NormalsLet’s do it…Let’s do it…Exploring How Normals Affect LightingLet’s do it…Summary
Technical requirementsTranslating points in 3DYour turn…Scaling points with x, y, and zLet’s do it…Your turn…Rotating points around a pivotLet’s do it…Exploring transformation ordersShearing and reflectionsSummaryAnswers
Technical requirementsDefining matricesPerforming operations on matricesAdding and subtracting matricesMultiplying by a single valueDividing matricesCreating matrix representations of affine transformationsMoving from linear equations to matrix operationsCompounding affine transformationsCombining transformation matrices for complex maneuversLet’s do it…SummaryAnswers
Technical requirementsUnderstanding OpenGL’s Matrix StackWorking with the Model MatrixLet’s do it…Your turn…Working with the View MatrixLet’s do it…Working with the Projection MatrixLet’s do it…Your turn…SummaryAnswers
Technical requirementsFlying maneuversLet’s do it…Your turn...Understanding and fixing compound rotation quirksImproving camera orientationsLet’s do it…SummaryAnswers
Technical requirementsIntroducing quaternionsRotating around an arbitrary axisExploring quaternion spacesLet’s do it…Working with unit quaternionsYour turn…Understanding the purpose of normalizationLet’s do it...Your turn...SummaryAnswers
Technical requirementsUnderstanding shadersTransferring processing from the CPU to the GPULet’s do it…Processing pixel by pixelLet’s do it…Summary
Technical requirementsColoring and texturing mesh facesLet’s do it…Let’s do it…Your turn...Turning on the lightsAmbient lightingDiffuse lightingPhongSummaryAnswers
Technical requirementsFollowing where light bouncesApplying the Inverse Square LawCalculating Bidirectional ReflectanceDistribution functionsThe Fresnel effectGeometric attentuation factorPutting it all togetherLet’s do it…Summary
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Content preview from Mathematics for Game Programming and Computer Graphics

2 Let’s Start Drawing

There’s nothing quite like getting your first graphics application running. There’s a real sense of achievement when you see your ideas for the rearrangements of pixels come together on the screen. My first coding language was BASIC on an Amstrad CPC664 and even though the code was quite laborious, I liked nothing better than to draw shapes and change the colors on the screen.

Computer-drawn images ultimately end up as single pixels (such as the ones drawn in the previous section) with differing colors on a graphics display. To draw all objects pixel by pixel whenever you’d like to create an image would be a long, drawn-out process. It might have been how images were rendered on a screen in the 1950s, but now, with advanced ...